Band mapping in higher-energy x-ray photoemission: Phonon effects and comparison to one-step theory

We have studied the temperature dependence of W(110) soft x-ray angle-resolved photoemission spectra excited at photon energies of 260 and 870 eV and between 300 and 780 K. The experimental results have been compared to both a free-electron final-state model and theoretical one-step model calculations of the photocurrent. At 300 K, clear band dispersions can be observed in the data. The temperature dependence of the data can be analyzed qualitatively in terms of a direct-transition band-dispersion regime (“UPS” limit) versus a nondirect-transition density-of-states regime (“XPS” limit). The ratio between direct and nondirect transitions is estimated from a Debye-Waller factor, which for example at $h\nu =870\text{eV}$ predicts 70% direct transitions at 300 K, and 41% at 780 K, and these values qualitatively describe our data. Beyond this, the state-of-the-art one-step theoretical calculations reproduce well the band dispersions and matrix element effects in the measured spectra at room temperature. However, simulating the temperature dependence is more complicated, and including phonon effects via complex phase shifts accounts for the suppression of existing direct-transition features, but does not reproduce new, density-of-states-related background intensity which shows up in higher-temperature experimental spectra. Finally, we also discuss the implications of this work for future experiments on other materials and at even higher photon energies up to 10 keV.

Figure 1

(Color online) The experimental geometry, with various key elements, and two key wave-vector conservation equations, indicated. The approximate curve sampled in our angle-resolved data is also shown. The angles $\theta$ and $\varphi$ here refer to those inside the surface, after allowance for slight refraction effects on $\theta$ due to the inner potential.

Figure 2

(Color online) (a)–(d) Plots of intensity versus angle of emission for 260 eV excitation from the valence bands of W(110) and for an emission angle that samples initial states roughly along the $\Gamma$-to-$N$ line in the Brillouin zone, and for four temperatures of 300, 470, 607, and 870 K. In (a) also the transitions allowed with free-electron final states are shown, solid lines are primary cone bands and the dashed lines surface umklapp bands with $\vec{\mathbf{g}}=⟨112⟩\pi /a$. The numbers label certain features that also appear in (e) and (f). (e) The temperature dependence of energy distribution curves as integrated over 20 channels in angle as indicated in as indicated in (a)-(d). A comparison to the $W$ density of states (DOS), as broadened by experimental resolution of 150 meV is also shown in the topmost curve. (f) The temperature dependence of momentum distribution curves at one selected energy, again as indicated in (a)-(d). (g) The average locus of points in the BZ sampled in this data, assuming direct transitions and free-electron final states.

Figure 3

(Color online) As in Fig. 2, but for one-step photoemission calculations including temperature effects via complex phase shifts. See text for details.

Figure 4

(Color online) As in Fig. 2 but for 870 eV excitation and an emission angle that approximately samples the $N$-to-$\Gamma$ line in the Brillouin zone. In (e) we show for comparison the $W$ density of states (DOS), as broadened by experimental resolution of 400 meV.

Figure 5

(Color online) As in Fig. 3 but for a choice of 890 eV excitation that was found to yield optimum agreement with experiment.

Figure 6

(Color online) Debye-Waller factors for valence-band photoemission from $W$ at various temperatures over 0–300 K and electron kinetic energies over 0–10 keV. These permit a rough estimate of the fraction of transitions yielding simple band mapping features.